Evaluation of the Scores Given by the Student Teachers

S

Step 5- Investigate the results of the reasoning by product-sum-gravity method

Compared with the figure 3.22 (conventional one), the figure 3.23 (improved one) is more natural for teachers because of the symmetry. Although their peaks are located at different points (^Ý

’b‰+³_’‹‰ and ³

Qb‰+^R_Q‹‰), both of these peaks can be considered as b_‰^X.

Applying the obtained reasoning results (figure 3.24 and figure 3.25) to compare the conventional (Zadeh’s) extension principle and the improved one again, we know that they both have the same supports [^R_·,^Ý_·] and both of their peaks are located at =³_Q. The graphs also show the symmetry of the two reasoning results; however, the former one is an isosceles triangular fuzzy number, and the latter one is in a well-blended shape (between an isosceles triangular fuzzy number and a quadratic-curved fuzzy number). For this reason, we still can say that the reasoning result obtained by the improved extension principle is the more appropriate one. Additionally, the improved extension principle is a more practical way to execute approximate reasoning on fuzzy numbers; unlike the cases applying Zadeh’s extension principle, we do not need to extend anything even if the supports of the input fuzzy numbers go beyond the grading scales.

F

Figure 3.26: The reasoning result (Taiwan / conventional)

Figure 3.27: The reasoning result (Taiwan / improved)

Along the membership grade, educators can divide the support of the reasoning result into some parts. Here, the reasoning result is divided into 4 parts:

á.â.= ãWäl(q,v)(W) ≥ 0.75å = k(, )_.݆

Y_á.â.= ãWä0.75 > _l(q,v)(W) ≥ 0.5å = k(, )_.†∩ k(, )_.݆

b_á.â.= ãWä0.5 > _l(q,v)(W) ≥ 0.25å = k(, )_.3†∩ k(, )_.†

‹_á.â.= ãWä0.25 > _l(q,v)(W) ≥ 0å = k(, )_.3†

For the reasoning result obtained by the conventional (Zadeh’s) extension principle and 0

0.25 0.5 0.75 1

-0.2 0 0.2 0.4 0.6 0.8 1

0 0.25 0.5 0.75 1

-0.2 0 0.2 0.4 0.6 0.8 1

Conventional extension principle Improved extension principle á.â.= [0.237,0.563]

(approx.)

á.â.= [0.266,0.568]

(approx.) Yá.â.= [0.139,0.237) ∪ (0.563,0.639]

(approx.)

Yá.â.= [0.167,0.266) ∪ (0.568,0.667]

(approx.) b_á.â.= [0.063,0.139) ∪ (0.639,0.737]

(approx.)

b_á.â. = [0.068,0.167) ∪ (0.667,0.767]

(approx.)

‹_á.â.= [0,0.063) ∪ (0.737,1]

(approx.)

‹_á.â.= [0,0.068) ∪ (0.767,1]

(approx.)

According to the given grades, _á.â., Y_á.â., b_á.â. and ‹_á.â., the educators can evaluate the grade given by the student teachers more easily. For example, if a Taiwanese student teacher ST-A grades the calligraphy work (figure 3.14) and gives it grade b ( =0.5 ), the educator (or the advisor) can grade this student teacher’s evaluation result as follows:

b( = 0.5 ) ∈ á.â.(¹ºE»¼E$&ºE9½)= [0.2365,0.5627](approx. ) and b( = 0.5 ) ∈ á.â.(&Þß%º»¼à)= [0.2658,0.5676](approx. ).

Based on the analysis results obtained above, this evaluation result b ( =0.5 ), given by student teacher ST-A, can obtain grade from the educator undoubtedly.

There is another student teacher ST-B grading the same calligraphy work (figure 3.14) but giving it grade Y^X (=³_QY +^R_Qb =³_Q×^Q+^R_Q×^R₃=³_Q), then the educator can grade ST-B’s evaluation result as follows:

Y^X( =³

Q ) ∈ bá.â.(¹ºE»¼E$&ºE9½)= [0.063,0.139) ∪ (0.639,0.737](approx. ) and Y^X( =³_Q ) ∈ Yá.â.(&Þß%º»¼à)= [0.167,0.266) ∪ (0.568,0.667](approx. ).

Obviously, the choice of whether to apply the improved extension principle or the conventional one will affect the grade for student teacher ST-B here, and the ST-B will receive higher grade if the educator applies the improved extension principle.

Conclusion

There are two extension principles compared in this study; one is Zadeh’s extension principle, the other one is an improved extension principle. With the same inputs, these two extension principles output in different ways.

F

Figure 1: Comparison 1

(symmetric; triangular) (symmetric; quadratic-curved)

Zadeh’s extension principle Improved extension principle

Input

Output Output

(asymmetric) (symmetric; well-blended)

k(Y/, Yq / )vX

F

Figure 2: Comparison 2

Both of the extension principles were conducted with symmetric input. However, as the reasoning results observed above, Zadeh’s extension principle did not always output a symmetric reasoning result (see figure 1). Even though Zadeh’s extension principle output a symmetric reasoning result (see figure 2), the reasoning result was not well-blended between the input triangular fuzzy number and the input quadratic-curved fuzzy number.

Applying approximate reasoning to educational evaluation, it is natural for teachers to expect symmetric reasoning results if they input symmetric fuzzy numbers. Therefore, we need an extension principle which can ensure symmetric reasoning result especially when they input symmetric fuzzy numbers. As the reasoning results obtained above, we observed that the improved can ensure the symmetry of the output in both cases.

If we input two fuzzy numbers in different shapes, they also tend to expect the output in a

(symmetric; triangular) (symmetric; quadratic-curved)

Input

Output Output

Zadeh’s extension principle Improved extension principle

(symmetric; triangular) (symmetric; well-blended)

shape blended with the input fuzzy numbers. For example, if we input a triangular fuzzy number and a quadratic-curved fuzzy number, we will expect an output in a shape blended between a triangular fuzzy number and a quadratic-curved fuzzy number instead of an output in a shape of a triangular fuzzy number or of a quadratic-curved fuzzy number.

Considering the symmetry and the shape of the output, the improved extension principle gave more appropriate reasoning results than Zadeh’s extension principle did. It can be concluded that the improved extension principle can give teachers or educators, who want to apply approximate reasoning to educational evaluation, more appropriate reasoning results especially when they input fuzzy numbers into approximate reasoning.

Acknowledgements

I would like to express my deep gratitude to my thesis adviser, Professor Takenobu Takizawa, for his continuing guidance, support and encouragement during my Ph. D research at Waseda University.

I sincerely thank Professor Takashi Yokomori (Waseda University), Professor Kimio Watanabe (Waseda University), Professor Emeritus Hajime Yamashita (Waseda University) and Professor Shuya Kanagawa (Tokyo City University), who gave me valuable comments as my dissertation committee members.

I am especially grateful to Professor Takizawa and Professor Emeritus Yamashita for their detailed advice on my study. They also encouraged me to have the chance to attend conferences / symposiums. I got many helpful advices and suggestions from audience there.

I am also thankful to Professor Jiro Inaida (Nihon University), Principal Ei Tsuda (Kokugakuin High School), Assistant Professor Hiroaki Uesu (Waseda University), and Assistant Professor Kimiaki Shinkai (Tokyo Kasei Gakuin University) for their valuable comments and advices to my ideas. Without their meaningful advices, I would never have validated my research idea.

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